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Chapter 5: Problem 43

Describe identities that can be verified using the sum-to-product formulas.

Short Answer

Expert verified
Sum-to-product formulas allow us to re-write sums and differences of sine or cosine expressions as a product. They can verify identities involving sums or differences of sine or cosine functions, as well as simplify integrals and sums in calculus, change the form of a trigonometric expression, or prove other identities.

Step by step solution

01

Identification of Sum-to-Product Formulas

The sum-to-product formulas are as follows: 1. \(\sin(A) + \sin(B) = 2 \sin \left(\frac{A + B}{2} \right) \cos \left(\frac{A - B}{2} \right)\)2. \(\sin(A) - \sin(B) = 2 \cos \left(\frac{A + B}{2} \right) \sin \left(\frac{A - B}{2} \right)\)3. \(\cos(A) + \cos(B) = 2 \cos \left(\frac{A + B}{2} \right) \cos \left(\frac{A - B}{2} \right)\)4. \(\cos(A) - \cos(B) = -2 \sin \left(\frac{A + B}{2} \right) \sin \left(\frac{A - B}{2} \right)\)
02

Verifying identities using these formulas

Given an identity involving sums or differences of sine or cosine, it can be converted to a product using the sum-to-product formulas. For example, consider verifying the first sum-to-product formula:1. Start with the left-hand side (LHS) i.e., \(\sin(A) + \sin(B) = 2 \sin \left(\frac{A + B}{2} \right) \cos \left(\frac{A - B}{2} \right)\)2. Use the sum-to-product formula to write the LHS as a product: 2 \(\sin \left(\frac{A + B}{2} \right) \cos \left(\frac{A - B}{2} \right)\). Hence, both sides of the identity are verified.
03

Introduction to general applications

These formulas are often used to simplify integrals and sums in calculus, to change the form of a trigonometric expression, or to prove other identities. The basic idea is to always try to simplify the expression by converting the sum or difference into a product format.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Identities
Trigonometric identities serve as fundamental tools in simplifying and manipulating trigonometric expressions. They are equations involving trigonometric functions that are true for every value of the variable involved.
In the context of sum-to-product formulas, these identities enable us to transform sums or differences of sines and cosines into products.
This can often make solving complex trigonometric problems more manageable. Some identities like Pythagorean identities, angle sum and difference identities, and double-angle identities frequently appear in calculus and geometry, helping ease calculations and proofs.
Trigonometric Simplification
Simplifying trigonometric expressions is a crucial skill in virtually all math-related fields. Trigonometric simplification involves reducing complex trigonometric expressions to simpler forms.
This is often done using identities like sum-to-product formulas, which convert sums or differences into products, facilitating easier manipulation.
  • Example: Transforming \(\sin(A) + \sin(B)\) using the sum-to-product formula simplifies to \( 2 \sin \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)\).
  • Simplifying can reduce the number of steps in calculations and helps in solving equations or evaluating complex integrals.
Understanding how to simplify trigonometric expressions aids not only in solving textbook problems but also in higher-level applications such as signal processing and physics calculations.
Sine and Cosine Transformations
Transforming sine and cosine functions is an essential part of mastering trigonometry. These transformations usually involve using identities, such as sum-to-product and product-to-sum identities, to express trigonometric functions in alternative forms.
The sum-to-product formulas specifically help in converting sums and differences into products. This conversion is particularly helpful because products are often easier to integrate or differentiate.
  • Example 1: \(\sin(A) + \sin(B)\) can be rewritten as \(2 \sin \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)\), a product.
  • Example 2: \(\cos(A) - \cos(B)\) can transform into \(-2 \sin \left(\frac{A + B}{2}\right) \sin \left(\frac{A - B}{2}\right)\).
Mastering these transformations allows one to tackle a variety of problems, from simple algebraic manipulations to complex calculus challenges.

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Most popular questions from this chapter

In Exercises \(85-96,\) use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$\tan x=-3$$

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Use this information to solve Exercises \(129-130 .\) Our cycle of normal breathing takes place every 5 seconds. Velocity of air flow, y, measured in liters per second, after \(x\) seconds is modeled by $$ y=0.6 \sin \frac{2 \pi}{5} x $$ Velocity of air flow is positive when we inhale and negative when we exhale. Within each breathing cycle, when are we inhaling at a rate of 0.3 liter per second? Round to the nearest tenth of a second.
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